MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  icomnfordt Structured version   Unicode version

Theorem icomnfordt 19511
Description: An unbounded above open interval is open in the order topology of the extended reals. (Contributed by Mario Carneiro, 3-Sep-2015.)
Assertion
Ref Expression
icomnfordt  |-  ( -oo [,) A )  e.  (ordTop `  <_  )

Proof of Theorem icomnfordt
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2467 . . . . . . . . 9  |-  ran  (
x  e.  RR*  |->  ( x (,] +oo ) )  =  ran  ( x  e.  RR*  |->  ( x (,] +oo ) )
2 eqid 2467 . . . . . . . . 9  |-  ran  (
x  e.  RR*  |->  ( -oo [,) x ) )  =  ran  ( x  e. 
RR*  |->  ( -oo [,) x ) )
3 eqid 2467 . . . . . . . . 9  |-  ran  (,)  =  ran  (,)
41, 2, 3leordtval 19508 . . . . . . . 8  |-  (ordTop `  <_  )  =  ( topGen `  ( ( ran  (
x  e.  RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )  u.  ran  (,) )
)
5 letop 19501 . . . . . . . 8  |-  (ordTop `  <_  )  e.  Top
64, 5eqeltrri 2552 . . . . . . 7  |-  ( topGen `  ( ( ran  (
x  e.  RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )  u.  ran  (,) )
)  e.  Top
7 tgclb 19266 . . . . . . 7  |-  ( ( ( ran  ( x  e.  RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )  u.  ran  (,) )  e. 
TopBases  <-> 
( topGen `  ( ( ran  ( x  e.  RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )  u.  ran  (,) )
)  e.  Top )
86, 7mpbir 209 . . . . . 6  |-  ( ( ran  ( x  e. 
RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )  u.  ran  (,) )  e. 
TopBases
9 bastg 19262 . . . . . 6  |-  ( ( ( ran  ( x  e.  RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )  u.  ran  (,) )  e. 
TopBases  ->  ( ( ran  ( x  e.  RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )  u.  ran  (,) )  C_  ( topGen `  ( ( ran  ( x  e.  RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )  u.  ran  (,) )
) )
108, 9ax-mp 5 . . . . 5  |-  ( ( ran  ( x  e. 
RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )  u.  ran  (,) )  C_  ( topGen `  ( ( ran  ( x  e.  RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )  u.  ran  (,) )
)
1110, 4sseqtr4i 3537 . . . 4  |-  ( ( ran  ( x  e. 
RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )  u.  ran  (,) )  C_  (ordTop `  <_  )
12 ssun1 3667 . . . . 5  |-  ( ran  ( x  e.  RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) ) 
C_  ( ( ran  ( x  e.  RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )  u.  ran  (,) )
13 ssun2 3668 . . . . . 6  |-  ran  (
x  e.  RR*  |->  ( -oo [,) x ) )  C_  ( ran  ( x  e. 
RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )
14 eqid 2467 . . . . . . . 8  |-  ( -oo [,) A )  =  ( -oo [,) A )
15 oveq2 6292 . . . . . . . . . 10  |-  ( x  =  A  ->  ( -oo [,) x )  =  ( -oo [,) A
) )
1615eqeq2d 2481 . . . . . . . . 9  |-  ( x  =  A  ->  (
( -oo [,) A )  =  ( -oo [,) x )  <->  ( -oo [,) A )  =  ( -oo [,) A ) ) )
1716rspcev 3214 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  ( -oo [,) A )  =  ( -oo [,) A
) )  ->  E. x  e.  RR*  ( -oo [,) A )  =  ( -oo [,) x ) )
1814, 17mpan2 671 . . . . . . 7  |-  ( A  e.  RR*  ->  E. x  e.  RR*  ( -oo [,) A )  =  ( -oo [,) x ) )
19 eqid 2467 . . . . . . . 8  |-  ( x  e.  RR*  |->  ( -oo [,) x ) )  =  ( x  e.  RR*  |->  ( -oo [,) x ) )
20 ovex 6309 . . . . . . . 8  |-  ( -oo [,) x )  e.  _V
2119, 20elrnmpti 5253 . . . . . . 7  |-  ( ( -oo [,) A )  e.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) )  <->  E. x  e.  RR*  ( -oo [,) A )  =  ( -oo [,) x ) )
2218, 21sylibr 212 . . . . . 6  |-  ( A  e.  RR*  ->  ( -oo [,) A )  e.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )
2313, 22sseldi 3502 . . . . 5  |-  ( A  e.  RR*  ->  ( -oo [,) A )  e.  ( ran  ( x  e. 
RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) ) )
2412, 23sseldi 3502 . . . 4  |-  ( A  e.  RR*  ->  ( -oo [,) A )  e.  ( ( ran  ( x  e.  RR*  |->  ( x (,] +oo ) )  u.  ran  ( x  e.  RR*  |->  ( -oo [,) x ) ) )  u.  ran  (,) )
)
2511, 24sseldi 3502 . . 3  |-  ( A  e.  RR*  ->  ( -oo [,) A )  e.  (ordTop `  <_  ) )
2625adantl 466 . 2  |-  ( ( -oo  e.  RR*  /\  A  e.  RR* )  ->  ( -oo [,) A )  e.  (ordTop `  <_  ) )
27 df-ico 11535 . . . . . 6  |-  [,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <  y ) } )
2827ixxf 11539 . . . . 5  |-  [,) :
( RR*  X.  RR* ) --> ~P RR*
2928fdmi 5736 . . . 4  |-  dom  [,)  =  ( RR*  X.  RR* )
3029ndmov 6443 . . 3  |-  ( -.  ( -oo  e.  RR*  /\  A  e.  RR* )  ->  ( -oo [,) A
)  =  (/) )
31 0opn 19208 . . . 4  |-  ( (ordTop `  <_  )  e.  Top  -> 
(/)  e.  (ordTop `  <_  ) )
325, 31ax-mp 5 . . 3  |-  (/)  e.  (ordTop `  <_  )
3330, 32syl6eqel 2563 . 2  |-  ( -.  ( -oo  e.  RR*  /\  A  e.  RR* )  ->  ( -oo [,) A
)  e.  (ordTop `  <_  ) )
3426, 33pm2.61i 164 1  |-  ( -oo [,) A )  e.  (ordTop `  <_  )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 369    = wceq 1379    e. wcel 1767   E.wrex 2815    u. cun 3474    C_ wss 3476   (/)c0 3785   ~Pcpw 4010    |-> cmpt 4505    X. cxp 4997   ran crn 5000   ` cfv 5588  (class class class)co 6284   +oocpnf 9625   -oocmnf 9626   RR*cxr 9627    < clt 9628    <_ cle 9629   (,)cioo 11529   (,]cioc 11530   [,)cico 11531   topGenctg 14693  ordTopcordt 14754   Topctop 19189   TopBasesctb 19193
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fi 7871  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-ioo 11533  df-ioc 11534  df-ico 11535  df-icc 11536  df-topgen 14699  df-ordt 14756  df-ps 15687  df-tsr 15688  df-top 19194  df-bases 19196  df-topon 19197
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator